Here \(\Lambda \)  is the cosmological constant. It is clear that the
cosmological constant is coupled to the dilaton field in a very
nontrivial way. This type of the dilaton potential was introduced
for the first time by Gao and Zhang [1]}. They derived, by
applying a coordinates transformation which recast the solution in
the Schwarzchild coordinates system, the static dilaton black hole
solutions in the background of (A)dS universe. For this purpose,
they required the existence of the (A)dS dilaton black hole
solutions and extracted successfully the form of the dilaton
potential leading to (A)dS-like solutions. They also argued that
this type of derived potential can be obtained when a higher
dimensional theory is compactified to four dimensions, including
various supergravity models [2]}. In the absence of the
dilaton field the action (REF ) reduces to the action of
Einstein-Maxwell gravity with cosmological constant. Varying the
action (REF ) with respect to the gravitational field \(g_{\mu \nu }\) , the dilaton field \(\Phi \)  and the gauge field \(A_{\mu }\) ,
yields
\({R}_{\mu \nu }=2 \partial _{\mu }\Phi \partial _{\nu }\Phi +\frac{1}{2}g_{\mu \nu }V(\Phi )+2e^{-2\alpha \Phi }\left( F_{\mu \eta }F_{\nu }^{\text{ }\eta }-\frac{1}{4}g_{\mu \nu }F_{\lambda \eta }F^{\lambda \eta }\right) , \) 
\(\nabla ^{2}\Phi =\frac{1}{4}\frac{\partial V}{\partial \Phi }-\frac{\alpha }{2}e^{-2\alpha \Phi }F_{\lambda \eta }F^{\lambda \eta }, \) 
\(\partial _{\mu }\left( \sqrt{-g}e^{-2\alpha \Phi }F^{\mu \nu }\right) =0. \) 
